gdp_change

Based on the plot of the realization, the assuming the series has a constant, non-zero mean appears to be reasonable. The sample autocorrelations are positive and fall off quickly. The parzen window shows a dominate frequency at 0 with some evidence of frequencies at approximately 0.2, 0.3 and 0.45, which is confirmed by the periodogram. The ACF for the first and second half of the series do not appear to be the same, which is evidence against this data coming from a stationary process.

vals <- plotts.sample.wge(data$gdp_change)

acf(data$gdp_change[1:68])
acf(data$gdp_change[69:135])

unrate

Based on the plot of the realization, a constant mean assumption seems reasonable. The parzen window and periodogram show system frequencies near zero. The periodogram suggests that the peak may be just above 0, possibly between 0.014 and 0.022. The sample autocorrelations are positive and fall off by the 10th lag. The sample autocorrelations of the first and second half of the realization apepar to be consistent.

AIC suggests a model with AR order of 4 or 5 with a moving avarege component. However, an AR(3) fit to the realization appears to be sufficient to reduce the realization to white noise. Additionally, the fitted AR(3) as a root in the range suggested by the periodogram. Higher order models (4-5), keep this root and add factors with roots relatively far from the unit circle.

The true plots of the AR(3) fit appear to be consistent with the behavior of the sample plots.

Sample Plots

vals <- plotts.sample.wge(data$unrate)

acf(data$unrate[1:68])
acf(data$unrate[69:135])

Model Fits

An AR(3) is fit below. The plot shows the behavior of the realization and the result of differencing the realization with the AR(3) fit. The resulting difference appears to be consistent with white noise.

aic5.wge(data$unrate)

## ---------WORKING... PLEASE WAIT... 
## 
## 
## Five Smallest Values of  aic

##       p    q        aic
## 18    5    2  -2.876830
## 14    4    1  -2.816447
## 15    4    2  -2.803668
## 17    5    1  -2.802325
## 8     2    1  -2.787907

est <- est.arma.wge(data$unrate, p = 3)

## 
## Coefficients of Original polynomial:  
## 1.3238 -0.0346 -0.3272 
## 
## Factor                 Roots                Abs Recip    System Freq 
## 1-1.7468B+0.7735B^2    1.1292+-0.1335i      0.8795       0.0187
## 1+0.4230B             -2.3640               0.4230       0.5000
##   
## 

vals <- artrans.wge(data$unrate, phi.tr = est$phi)

True Plots of AR(3) Fit

The true characteristics of the AR(3) fit is plotted below.

vals <- plotts.true.wge(135, phi = est$phi)

nfjobs

The realization appears to wander with a upward trend. The periodgram and parzen window show evidence of a dominate peak at zero. The sample autocorrelations are positive and fall off slowly. These sample plots appear to be consistent with a non-stationary time series with an upward trend.

With the first difference taken, the result appears to be consistent with an ARMA process. AIC suggests a noise structure of AR(3). An AR(3) model fit appears to be sufficient to reduce the noise structure to white noise.

An ARIMA(3,1,0) may be sufficient to model this realization.

Sample Plots

vals <- plotts.sample.wge(data$nfjobs)

acf(data$nfjobs[1:68])
acf(data$nfjobs[69:135])

Model Fit

When the first difference is taken from the realization, the noise structure appears to be consistent with an ARMA model. AIC suggests an AR(3). When the fitted AR(3) is differenced with the noise, the result appears to be white noise.

nfjobs.first.diff <- artrans.wge(data$nfjobs, phi.tr = c(1))

plotts.sample.wge(nfjobs.first.diff)

## $autplt
##  [1]  1.00000000  0.85810060  0.73629599  0.58402645  0.43744852
##  [6]  0.31321523  0.20679470  0.11457280 -0.00388365 -0.06557015
## [11] -0.12720345 -0.17876145 -0.21910103 -0.22372587 -0.23481412
## [16] -0.23406263 -0.21215105 -0.17531641 -0.14841574 -0.12716410
## [21] -0.08744396 -0.06921297 -0.01794801  0.01872713  0.06919132
## [26]  0.07268053
## 
## $freq
##  [1] 0.007462687 0.014925373 0.022388060 0.029850746 0.037313433
##  [6] 0.044776119 0.052238806 0.059701493 0.067164179 0.074626866
## [11] 0.082089552 0.089552239 0.097014925 0.104477612 0.111940299
## [16] 0.119402985 0.126865672 0.134328358 0.141791045 0.149253731
## [21] 0.156716418 0.164179104 0.171641791 0.179104478 0.186567164
## [26] 0.194029851 0.201492537 0.208955224 0.216417910 0.223880597
## [31] 0.231343284 0.238805970 0.246268657 0.253731343 0.261194030
## [36] 0.268656716 0.276119403 0.283582090 0.291044776 0.298507463
## [41] 0.305970149 0.313432836 0.320895522 0.328358209 0.335820896
## [46] 0.343283582 0.350746269 0.358208955 0.365671642 0.373134328
## [51] 0.380597015 0.388059701 0.395522388 0.402985075 0.410447761
## [56] 0.417910448 0.425373134 0.432835821 0.440298507 0.447761194
## [61] 0.455223881 0.462686567 0.470149254 0.477611940 0.485074627
## [66] 0.492537313 0.500000000
## 
## $db
##  [1]   8.7476425   8.9868580   3.1725841  11.5370117   8.0523554
##  [6]  10.0507440  -0.3228851   1.9338724   4.8084750   3.1331870
## [11]  -5.9824275  -3.0208779   2.8417081  -7.7383477  -0.8460293
## [16]  -8.0168360   0.5720412  -5.5467975  -4.4893818  -7.0137186
## [21] -16.9859326  -0.9124583 -10.3624216  -3.7877375 -14.1451992
## [26] -17.9362678  -8.9260476  -6.0823879  -9.4604740 -12.1017353
## [31] -10.3379745 -20.6139095  -8.2193690 -19.6481657 -21.3291214
## [36] -16.8714519 -14.8433161  -6.7467285  -7.0286750  -8.2198747
## [41]  -7.2954192 -12.5799551  -6.2836849 -24.1774663 -15.1657900
## [46] -11.8136606  -8.3224166 -11.8111798 -12.6084616 -11.7167764
## [51]  -9.6473866 -22.1276546 -13.0616227  -9.2519105  -6.6765530
## [56] -22.6533088  -7.3819811  -6.7064706 -14.9933078 -17.4827660
## [61]  -3.7554578 -11.0479278 -13.6701246 -14.5680753 -14.1570193
## [66] -22.2554324  -8.0939871
## 
## $dbz
##  [1]   7.9902135   8.0257639   8.0253019   7.9266867   7.6767347
##  [6]   7.2398704   6.6002911   5.7626519   4.7540027   3.6264436
## [11]   2.4559229   1.3294194   0.3177611  -0.5510740  -1.2938684
## [16]  -1.9475261  -2.5427838  -3.0954815  -3.6134068  -4.1065323
## [21]  -4.5912270  -5.0869614  -5.6098336  -6.1676098  -6.7582899
## [26]  -7.3721455  -7.9964693  -8.6213838  -9.2431032  -9.8595319
## [31] -10.4549137 -10.9774053 -11.3280076 -11.3945497 -11.1358823
## [36] -10.6341171 -10.0444317  -9.5095123  -9.1203243  -8.9191977
## [41]  -8.9141809  -9.0899871  -9.4138217  -9.8384040 -10.3050569
## [46] -10.7492642 -11.1096732 -11.3391461 -11.4142431 -11.3389954
## [51] -11.1407325 -10.8595510 -10.5364665 -10.2054966  -9.8918968
## [56]  -9.6152274  -9.3940159  -9.2488615  -9.2023724  -9.2761677
## [61]  -9.4860553  -9.8359962 -10.3103193 -10.8633609 -11.4089790
## [66] -11.8226486 -11.9789628

aic5.wge(nfjobs.first.diff)

## ---------WORKING... PLEASE WAIT... 
## 
## 
## Five Smallest Values of  aic

##       p    q        aic
## 10    3    0   11.27045
## 11    3    1   11.27502
## 6     1    2   11.27528
## 4     1    0   11.27583
## 13    4    0   11.27819

nfjobs.diff.model <- est.arma.wge(nfjobs.first.diff, p = 3)

## 
## Coefficients of Original polynomial:  
## 0.8588 0.1517 -0.1827 
## 
## Factor                 Roots                Abs Recip    System Freq 
## 1-0.7104B              1.4076               0.7104       0.0000
## 1-0.5867B              1.7046               0.5867       0.0000
## 1+0.4383B             -2.2817               0.4383       0.5000
##   
## 

vals <- artrans.wge(nfjobs.first.diff, phi.tr = nfjobs.diff.model$phi)

treas10yr

This realization appears to trend downward. A constant mean assumption does not appear to be reasonable for this realization. The sample autocorrelations are positive and drop off slowly. The parzen window and periodogram show evidence of a root at 0. These plots are consistent with a non-stationary process.

An ARIMA(0,1,0) appears to be sufficient to model this realization. When the first difference is taken, the result appears to be consistent with white noise.

Sample Plots

vals <- plotts.sample.wge(data$treas10yr)

acf(data$treas10yr[1:68])
acf(data$treas10yr[69:135])

Model Fit

When the first difference is taken from the realization, the noise structure appears to be consistent with an white noise.

treas10yr.first.diff <- artrans.wge(data$treas10yr, phi.tr = c(1))

plotts.sample.wge(treas10yr.first.diff)

## $autplt
##  [1]  1.000000000 -0.026363404 -0.165055285  0.003719405 -0.060939992
##  [6] -0.142350676  0.042998456 -0.090287894 -0.024491516 -0.008916625
## [11]  0.050526098  0.031668904 -0.002427237 -0.077783289 -0.007397616
## [16]  0.045576501  0.011156800 -0.048834230 -0.087246570  0.046812969
## [21]  0.070457067 -0.068400513  0.014399020  0.008625057  0.014067922
## [26] -0.041722208
## 
## $freq
##  [1] 0.007462687 0.014925373 0.022388060 0.029850746 0.037313433
##  [6] 0.044776119 0.052238806 0.059701493 0.067164179 0.074626866
## [11] 0.082089552 0.089552239 0.097014925 0.104477612 0.111940299
## [16] 0.119402985 0.126865672 0.134328358 0.141791045 0.149253731
## [21] 0.156716418 0.164179104 0.171641791 0.179104478 0.186567164
## [26] 0.194029851 0.201492537 0.208955224 0.216417910 0.223880597
## [31] 0.231343284 0.238805970 0.246268657 0.253731343 0.261194030
## [36] 0.268656716 0.276119403 0.283582090 0.291044776 0.298507463
## [41] 0.305970149 0.313432836 0.320895522 0.328358209 0.335820896
## [46] 0.343283582 0.350746269 0.358208955 0.365671642 0.373134328
## [51] 0.380597015 0.388059701 0.395522388 0.402985075 0.410447761
## [56] 0.417910448 0.425373134 0.432835821 0.440298507 0.447761194
## [61] 0.455223881 0.462686567 0.470149254 0.477611940 0.485074627
## [66] 0.492537313 0.500000000
## 
## $db
##  [1]  -9.39317070  -7.07830109  -7.84556073  -3.66611542  -7.68108430
##  [6]   0.10652004  -3.60713218  -3.86646586   1.08005163   2.79722864
## [11]   0.08835111  -0.05789957  -0.56694372   5.29883771  -2.99163502
## [16]  -9.67703391  -2.52950382   4.79508341   0.06189730  -7.41599010
## [21]  -8.50404512  -6.26854479   2.43892715   0.95706555  -1.62411128
## [26]   2.70693192   4.47416574  -8.97306379   2.42025111  -4.36719546
## [31]  -5.51345317  -3.45335077  -0.35563751   4.83268444  -3.58146323
## [36]  -4.37209072   4.92339562  -6.73625507   1.53700451 -12.14495473
## [41]   5.79433583  -2.56505205   4.51386730  -2.98450543  -5.89198755
## [46]   3.55493734   4.19064038  -5.70279446   3.52334806  -6.82702707
## [51]  -3.22510897   2.34555577  -9.62330209  -1.40218484  -1.09106758
## [56]   0.61385229  -6.30621533  -7.45689407  -7.61038596  -2.44294198
## [61]  -0.57838218  -1.86113242   0.23178254  -2.83173326  -0.45390143
## [66]   3.48646521 -11.77063934
## 
## $dbz
##  [1] -5.66123742 -5.07066537 -4.29166592 -3.47336101 -2.69656096
##  [6] -1.99073679 -1.36062897 -0.80479378 -0.32452158  0.07453474
## [11]  0.38488663  0.60091612  0.72176264  0.75264886  0.70488366
## [16]  0.59534035  0.44632673  0.28616522  0.14919001  0.07192225
## [21]  0.08257579  0.18663903  0.35857401  0.54811480  0.69797767
## [26]  0.76238410  0.71983694  0.58009290  0.38594628  0.20594140
## [31]  0.11181368  0.14506418  0.29628412  0.51706386  0.75205280
## [36]  0.96377988  1.13908987  1.28240598  1.40431595  1.51183082
## [41]  1.60390195  1.67242637  1.70605076  1.69355230  1.62531900
## [46]  1.49350684  1.29231293  1.01934594  0.67796955  0.27947072
## [51] -0.15657886 -0.60498161 -1.04010028 -1.44051885 -1.78906350
## [56] -2.06712993 -2.24832551 -2.30015518 -2.19826329 -1.94529799
## [61] -1.57778797 -1.15415361 -0.73484478 -0.36856844 -0.08837120
## [66]  0.08620861  0.14537603

treas3mo

This realization appears to be consistent with a downward trend with wandering or psuedo-periodic behavior. The autocorrelations are positive and decrease relatively quickly was the lags increase. However, the autocorrelations do not go to zero, which could suggest a non-stationary process. The ACF for the first and second half of the realization appear to be similar. When the ACF is calculated on half of the realization, the ACF apprears as damped exponentials in a sinusiodal envolope. This is typically evidence of an AR process with a complex root.

Since there appears to be a downward trend, a linear model was fit and the downward trend was removed (however, we would not expect a negative treasury yield). The resulting series appears to be consistent with an AR process with a complex root - pseudo-periodic behavior is present and the ACF are damped exponentials in a sinusiodal envolope. An ARMA(2,1) was suggested by AIC for the noise structure. When the AR(2) portion of the ARMA(2,1) fit to the noise is removed from the noise, the results is consistent with an MA(1).

Sample Plots

vals <- plotts.sample.wge(data$treas3mo)

acf(data$treas3mo[1:68])
acf(data$treas3mo[69:135])

Model Fit

When a linear model is fit, the resulting noise structure appear to be consistent with an ARMA process with a complex AR root. A ARMA(2,1) was suggested by AIC. When the AR(2) portion of the model is removed, the resulting noise appears to be consistent with an MA(1).

lin.model <- lm(data$treas3mo ~ seq_along(data$treas3mo))
plotts.sample.wge(lin.model$residuals)

## $autplt
##  [1]  1.00000000  0.94023639  0.84652083  0.71732832  0.56276237
##  [6]  0.41038248  0.25267305  0.09212126 -0.05266169 -0.18114794
## [11] -0.28604745 -0.36688910 -0.42758644 -0.46694966 -0.48638861
## [16] -0.47573985 -0.44315300 -0.39379744 -0.32851079 -0.25978751
## [21] -0.19210441 -0.12103369 -0.06230445 -0.01407123  0.01420941
## [26]  0.02216446
## 
## $freq
##  [1] 0.007407407 0.014814815 0.022222222 0.029629630 0.037037037
##  [6] 0.044444444 0.051851852 0.059259259 0.066666667 0.074074074
## [11] 0.081481481 0.088888889 0.096296296 0.103703704 0.111111111
## [16] 0.118518519 0.125925926 0.133333333 0.140740741 0.148148148
## [21] 0.155555556 0.162962963 0.170370370 0.177777778 0.185185185
## [26] 0.192592593 0.200000000 0.207407407 0.214814815 0.222222222
## [31] 0.229629630 0.237037037 0.244444444 0.251851852 0.259259259
## [36] 0.266666667 0.274074074 0.281481481 0.288888889 0.296296296
## [41] 0.303703704 0.311111111 0.318518519 0.325925926 0.333333333
## [46] 0.340740741 0.348148148 0.355555556 0.362962963 0.370370370
## [51] 0.377777778 0.385185185 0.392592593 0.400000000 0.407407407
## [56] 0.414814815 0.422222222 0.429629630 0.437037037 0.444444444
## [61] 0.451851852 0.459259259 0.466666667 0.474074074 0.481481481
## [66] 0.488888889 0.496296296
## 
## $db
##  [1]   4.3563517   7.8090559   9.9545026  11.9354468  -0.3463121
##  [6]  13.4962287   3.4245982   1.3124595   0.1017819 -13.8918744
## [11]   2.4565016  -8.7823000  -3.9731113  -4.0092396  -7.6379370
## [16] -15.4633870  -5.6954674  -4.5703892 -12.6192615  -7.4808032
## [21] -11.2836926  -5.9141248  -8.7307152 -21.1063666 -13.6744395
## [26] -15.6077732 -23.5796632 -16.2358171 -16.1953634 -13.0611279
## [31] -15.6806298 -28.2177787 -17.3586739 -18.4965037 -17.8755540
## [36] -13.3565919 -16.1936094 -24.9556558 -16.2891909 -17.2919148
## [41] -17.8774729 -14.6784880 -12.8586942 -15.1493330 -20.0390169
## [46] -25.3905758 -13.2947523 -17.5450977 -18.4937626 -14.8639111
## [51] -17.0486229  -9.1325544 -23.1136882 -12.4291670 -21.9831063
## [56] -19.9834425 -15.8073182 -13.6892978 -21.6263077 -15.9296655
## [61] -16.0134040 -22.9690451 -11.4678710 -18.7022789 -17.1846532
## [66] -21.3855547 -18.1339743
## 
## $dbz
##  [1]   8.263770614   8.447493768   8.626431872   8.685070665   8.537012777
##  [6]   8.128509458   7.430709664   6.433109873   5.142576374   3.589198977
## [11]   1.838493496   0.003120947  -1.767215391  -3.341163041  -4.673960070
## [16]  -5.800914887  -6.756772265  -7.527131564  -8.086294201  -8.465535892
## [21]  -8.762174646  -9.086652325  -9.515814462 -10.082872915 -10.788273209
## [26] -11.612213177 -12.519796240 -13.458010897 -14.352312327 -15.118559269
## [31] -15.698740437 -16.095930850 -16.367429474 -16.578959025 -16.762910086
## [36] -16.905556005 -16.960446403 -16.878165045 -16.638713179 -16.267231898
## [41] -15.823298127 -15.375356745 -14.978846501 -14.665964480 -14.445142787
## [46] -14.306553663 -14.231449534 -14.203208138 -14.216275527 -14.278759888
## [51] -14.407486961 -14.618582974 -14.918037472 -15.295056781 -15.719432085
## [56] -16.144373730 -16.516797027 -16.794311136 -16.961331564 -17.033293728
## [61] -17.045853192 -17.037623853 -17.036807300 -17.055526081 -17.090561457
## [66] -17.128360647 -17.152674153

aic5.wge(lin.model$residuals)

## ---------WORKING... PLEASE WAIT... 
## 
## 
## Five Smallest Values of  aic

##       p    q        aic
## 8     2    1  -1.752957
## 13    4    0  -1.751927
## 15    4    2  -1.747133
## 14    4    1  -1.746633
## 11    3    1  -1.745400

est <- est.arma.wge(lin.model$residuals, p = 2, q = 1)

## 
## Coefficients of Original polynomial:  
## 1.8714 -0.9154 
## 
## Factor                 Roots                Abs Recip    System Freq 
## 1-1.8714B+0.9154B^2    1.0222+-0.2182i      0.9568       0.0335
##   
## 

artrans.wge(lin.model$residuals, phi.tr = est$phi)

##   [1] -0.4460281291  1.1585323922 -0.5192686955  0.1264657907  0.7876202705
##   [6] -1.7918597174  0.8778718882  0.8735581334 -0.0791880747  0.2796160925
##  [11]  0.1542292026 -1.6172667179  0.9075410678 -0.2083491634  0.7018756787
##  [16] -0.2189106934 -0.4684223722 -0.0912004390  0.0092501486  0.4528658920
##  [21] -0.2517426969 -0.8995233906  1.3136672479 -0.7295745634 -0.5176278338
##  [26]  1.1112439611 -0.6588272728  0.2339410811 -0.3475091333  0.1067001724
##  [31]  0.3207640670  0.1885123373 -0.1305688149  0.3820029999 -0.5681553496
##  [36] -0.4145454305  0.0948533225 -0.0848482821  0.3258191215  0.0473939055
##  [41] -0.0563777509  0.1365820562  0.1113051603 -0.1842741398 -0.0566462538
##  [46]  0.5097098763 -0.4232922407  0.1679364134 -0.6305255792  0.7929875723
##  [51] -0.0687681515  0.3055468513 -0.1256621413  0.4050007975  0.2067579372
##  [56] -0.4082332222  0.4475701844 -0.5450751272 -1.1760844754  0.8404978533
##  [61] -0.6462562006  0.4402864141  0.5834810644 -0.2019145205 -0.1153860891
##  [66] -0.3021583381  0.1561554796 -0.2521886130  0.1760440813 -0.1348847646
##  [71] -0.0867798568  0.2955552011 -0.0332312851  0.1158330879  0.0816190380
##  [76] -0.1758757513  0.1319689506  0.1895625443  0.1345369246 -0.0272224751
##  [81] -0.3525406124  0.3522100027  0.0214189979 -0.1146717842 -0.6823268398
##  [86]  0.5299933629 -1.5022372343  2.3043327236 -1.4589319661  0.0433545653
##  [91]  0.7644122776 -0.1818614811 -0.1005592738 -0.1029635317  0.1033114751
##  [96] -0.1347224753 -0.0982751673 -0.0802047198 -0.0513227025 -0.0894623500
## [101] -0.0123068530 -0.0461818902 -0.0030008023 -0.0742343686 -0.0535569106
## [106] -0.1016276376  0.0234300145 -0.0874322272 -0.0306480548  0.0316525428
## [111] -0.0978889635 -0.0223566060 -0.0396157633  0.0109927743 -0.0524075575
## [116] -0.0330508348 -0.0124422972  0.1502985456 -0.1059399640 -0.0007100107
## [121] -0.0161737272  0.1957900061  0.0638850273  0.0697643036 -0.1743219598
## [126]  0.3490291129  0.1012731609 -0.0305774166  0.1687181237  0.1275757352
## [131] -0.1686428334 -0.1147359319  0.1258135717

treas10yr3mo

This realization appears to be consistent with an AR process with a complex root.

Sample Plots

vals <- plotts.sample.wge(data$treas10yr3mo)

acf(data$treas10yr3mo[1:68])
acf(data$treas10yr3mo[69:135])

Model Fit

A AR(3) model is suggested by AIC. A AR(3) appears to be sufficient to model is realization. When the fitted AR(3) is removed from the realization, the result appears to be consistent with white noise.

aic5.wge(data$treas10yr3mo)

## ---------WORKING... PLEASE WAIT... 
## 
## 
## Five Smallest Values of  aic

##       p    q        aic
## 10    3    0  -1.453501
## 13    4    0  -1.449416
## 7     2    0  -1.437387
## 14    4    1  -1.436171
## 16    5    0  -1.436059

est <- est.arma.wge(data$treas10yr3mo, p = 3)

## 
## Coefficients of Original polynomial:  
## 1.0769 -0.0485 -0.1704 
## 
## Factor                 Roots                Abs Recip    System Freq 
## 1-1.4080B+0.5147B^2    1.3677+-0.2687i      0.7174       0.0309
## 1+0.3311B             -3.0203               0.3311       0.5000
##   
## 

artrans.wge(data$treas10yr3mo, phi.tr = est$phi)

##   [1] -0.666398332  0.522725064  1.045340144  0.462228227  0.367803111
##   [6]  0.058569890 -0.236187036 -0.190361223 -0.156428156 -0.342837267
##  [11]  0.037175150  0.436919218 -0.005828953  0.419050906 -0.166114986
##  [16]  1.037257127  0.008266359  0.712213810  0.553776493 -0.163961189
##  [21]  0.855936175  0.967066256  0.349551742  0.494957895  0.398888788
##  [26]  0.027621287  0.172503127  0.266967496  0.809946180  0.815462411
##  [31]  0.169660488  0.129438242 -0.180198781 -0.344245020 -0.226026662
##  [36]  0.545294445 -0.094619590  0.834438290  0.401519989  0.051399127
##  [41] -0.199996467  0.597072804 -0.102190466 -0.012725733 -0.422753623
##  [46]  0.356205495 -0.009623543 -0.204913843  0.198037490  0.638274168
##  [51]  0.231766286 -0.023305928  0.201116336 -0.831041770  0.216663827
##  [56] -0.393377650 -0.274105491  1.463874460  0.980920885  0.193311260
##  [61]  1.154184295  0.452481716 -0.212446243 -0.599184496  1.163675764
##  [66]  0.517910677  0.220996328  0.742453709  0.665221521 -0.069166926
##  [71]  0.830449149 -0.405829294  0.058844543  0.213373382 -0.519262160
##  [76]  0.344993757 -0.209984999  0.072559614  0.042001093 -0.336766238
##  [81]  0.005207628 -0.044448971  0.572323696  0.472098299 -0.205454045
##  [86]  1.410897865  0.025126882  0.895721998 -0.560986912  0.693915560
##  [91]  1.251051986  0.059330488  0.964521593  0.321768590 -0.448653455
##  [96]  0.190074612  1.390409969  0.546091341  0.068459661 -0.786113835
## [101]  0.552883181  0.775317767 -0.331453587  0.272090090  0.505664785
## [106]  0.281529763  0.889773983  0.331581459  0.575757229  0.031536948
## [111]  0.194674558  0.454848844  0.015435822  0.161980198  0.812634285
## [116] -0.004150451  0.330750437 -0.203403065 -0.007188435  0.421254114
## [121]  0.856574766 -0.175907793 -0.168635666  0.301837371 -0.015994367
## [126]  0.202152993  0.097828472  0.090436765 -0.469320106 -0.049916066
## [131]  0.027442299 -0.029390348

fedintrate

The behavior of this realization appear to be correlated to treas3mo. The scatter plot of fedintrate vs treas3mo shows that there is a strong correlation between the two variables, \(\rho = 0.994\)

Sample Plots

vals <- plotts.sample.wge(data$fedintrate)

acf(data$fedintrate[1:68])
acf(data$fedintrate[69:135])

lm.fedtreas3mo <- lm(data$fedintrate ~ data$treas3mo)
plot(data$fedintrate, data$treas3mo)
cor(data$fedintrate, data$treas3mo)

## [1] 0.9942307

libor3mo

Like fedintrate, libor3mo appears to be highly correlated to treas3mo, \(\rho = 0.990\).

Sample Plots

vals <- plotts.sample.wge(data$libor3mo)

acf(data$libor3mo[1:68])
acf(data$libor3mo[69:135])

lm.fedtreas3mo <- lm(data$libor3mo ~ data$treas3mo)
plot(data$libor3mo, data$treas3mo)
cor(data$libor3mo, data$treas3mo)

## [1] 0.9909141

personincomechg

The sample plots seem to suggest an ARMA process with more than one complex autoregressive root. BIC suggests a model order of ARMA(5,0). When a fitted AR(5) model is removed from the realization, the result appears to be consistent with white noise.

Sample Plots

vals <- plotts.sample.wge(data$personincomechg)

acf(data$personincomechg[1:68])
acf(data$personincomechg[69:135])

Model Fit

A AR(5) model is suggested by BIC. When the fitted AR(5) is removed from the realization, the result appears to be consistent with white noise.

aic5.wge(data$personincomechg, p = 0:10, type = 'bic')

## ---------WORKING... PLEASE WAIT... 
## 
## 
## Five Smallest Values of  bic

##       p    q        bic
## 16    5    0 0.09867278
## 26    8    1 0.11675617
## 19    6    0 0.12730498
## 17    5    1 0.12959441
## 14    4    1 0.14310161

est <- est.arma.wge(data$personincomechg, p = 5)

## 
## Coefficients of Original polynomial:  
## 0.7277 0.2854 -0.1482 -0.5545 0.3989 
## 
## Factor                 Roots                Abs Recip    System Freq 
## 1+1.3228B+0.7502B^2   -0.8817+-0.7454i      0.8661       0.3883
## 1-1.2812B+0.6912B^2    0.9267+-0.7667i      0.8314       0.1100
## 1-0.7694B              1.2998               0.7694       0.0000
##   
## 

artrans.wge(data$personincomechg, phi.tr = est$phi)

##   [1] -1.451804495  2.048849096  2.063493628  1.327462898  1.751142526
##   [6]  0.817123471  0.788239490  1.101312888  0.998199367  0.362312173
##  [11]  0.929584381  0.433106964  1.147037149  0.170927206 -1.300083281
##  [16] -0.396915947  1.062378644  0.566076964  1.237554151  2.087248839
##  [21]  0.850980340  0.692712124  1.184094064  0.110953298  0.225290130
##  [26]  0.696307070  0.973547730  0.282625165  0.909487062  0.916400728
##  [31]  1.423157481  1.121465763  0.418707242  1.272745495  0.672487575
##  [36]  0.996954666  1.329019307  1.090898298  0.344011472  1.008828854
##  [41]  1.142563266  1.277170521  2.013887283  2.452547221  1.757856844
##  [46]  1.261427998  1.137008527  0.744004006  0.558664233  0.916034757
##  [51]  1.820038538  1.747451550  1.540867259  1.436149053  0.366799161
##  [56]  0.466062803  0.410400346  2.482162596 -0.560364064  1.321681012
##  [61]  1.573376315 -0.961115848  1.295232635 -0.010062371  1.200688680
##  [66]  1.755125578  1.573758644 -0.182268672  1.050751811  0.637319942
##  [71]  1.670927142 -0.204556961  0.180515943  0.004171037  1.149154077
##  [76]  3.052236737  0.531572562 -0.014207333  0.819330645  1.389899404
##  [81]  0.525662127  0.918822574 -0.298642788 -0.578228083  2.210034202
##  [86] -1.371808366 -0.029144734 -0.271824152  0.094957720  0.009090789
##  [91] -0.161470317  0.397094408  0.465527815  2.416656008  0.741546003
##  [96]  1.250323229 -1.024896591  0.858013177  1.067325807  1.854792688
## [101]  1.232278986 -0.507273508  2.650879617 -4.224774051 -0.430751525
## [106]  1.709155295 -0.401852271  2.063564863  1.858553934  0.961372715
## [111]  0.208912853  3.181721375  0.627518622  1.131576469  0.772655878
## [116]  0.540848978 -0.077542326  0.329362850  0.635519337  1.091146502
## [121]  0.924896295  0.681433468  1.011067689  1.552412960  1.281350076
## [126]  1.214761366  1.029866748  0.675889717  0.699761915  1.070553277

corpprofitchg

The sample plots appear to be consistent with white noise. Additionally, the model chosen by AIC is an ARMA(0,0), which is consistent with white noise.

Sample Plots

vals <- plotts.sample.wge(data$corpprofitchg)

acf(data$corpprofitchg[1:68])
acf(data$corpprofitchg[69:135])
aic5.wge(data$corpprofitchg, p = 0:5)

## ---------WORKING... PLEASE WAIT... 
## 
## 
## Five Smallest Values of  aic

##      p    q        aic
## 1    0    0   4.092074
## 4    1    0   4.094418
## 2    0    1   4.095979
## 3    0    2   4.103274
## 7    2    0   4.105762

GDP Change vs Unemployment Rate

There is evidence of a weak correlation between gdp_change and the first 10 lags of unrate.

plot.cross.lags(data, 'gdp_change', 'unrate', 19)

GDP Change vs Non-farm Jobs

There is evidence of a weak correlation between gdp_change and the first 10 lags of nfjobs.

plot.cross.lags(data, 'gdp_change', 'nfjobs', 19)

GDP Change vs 10 Year Treasury Yield

There is evidence of a weak correlation between gdp_change and the first 10 lags of treas10yr.

plot.cross.lags(data, 'gdp_change', 'treas10yr', 19)

GDP Change vs 3 Month Treasury Yield

There is evidence of a weak correlation between gdp_change and the first 10 lags of treas3mo.

plot.cross.lags(data, 'gdp_change', 'treas3mo', 19)

GDP Change vs Treasury Yield 10 Year 3 Month Difference

There is evidence of a weak correlation between gdp_change and the 2nd and 8th lags of treas10yr3mo.

plot.cross.lags(data, 'gdp_change', 'treas10yr3mo', 19)

GDP Change vs Federal Interest Rate

There is evidence of a weak correlation between gdp_change and the first 10 lags of fedintrate.

plot.cross.lags(data, 'gdp_change', 'fedintrate', 19)

GDP Change vs 3 Month LIBOR

There is evidence of a weak correlation between gdp_change and the first 10 lags of libor3mo.

plot.cross.lags(data, 'gdp_change', 'libor3mo', 19)

GDP Change vs Change in Personal Income

There is evidence of a weak correlation between gdp_change and the first 4 lags of personincomechg.

plot.cross.lags(data, 'gdp_change', 'personincomechg', 19)

GDP Change vs Change in Corparate Profits

There does not seem to be a correlation between change in GDP and lags of change in corparate profits.

plot.cross.lags(data, 'gdp_change', 'corpprofitchg', 19)

Explanatory Variables

There is strong evidence of linear relationships between fedintrate, libor3mo, and treas3mo. corpprofitchg does not seem to be correlated to anything.

data %>%
  select(-c('date', 'gdp_change')) %>%
  ggpairs()